There is not a chance this will happen without the aid of external tools and those are well beyond the scope of GZDoom, itself. You may want to explore our Software and Ports forum and the Editors / Asset Manipulation forum for examples of how tools are typically made to work seamlessly with GZDoom, itself.

With the abundance of languages and programming platforms available it should not be hard to find someone who's willing to integrate with the mod.io API and do what you are asking, here.

I'm having a heck of a time trying to find any docs on mod.io's site about what the experience is from a modder/user side. Everything seems to be geared towards "hey, use our API!" without much explanation on how to do things.

From a cursory glance, it looks like it's mostly trying to follow the Steam Workshop model of "subscribing" to a mod -- i.e. the user hits the button and it gets auto-applied to the game on launch, which isn't really a great fit for how Doom mods work.

It does look like there's a separate "download" button though, and there's built-in version management, so that's something. Maybe.

Also, maybe I'm wrong, but this seems designed for game where you can slap a bunch of mods together and just let the game run them all. ZDoom ... doesn't exactly do that, not outside of mod combinations that either don't conflict the slightest bit or have been specifically designed and coded to have forced compatibility.